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1 Real-tme Dscete Nonlnea Identfcaton va Recuent Hgh Ode Neual Netwoks Identfcacón No Lneal en empo Real usando Redes Neuonales Recuentes de Alto Oden Alma Y. Alans 1, Edga N. Sanchez and Alexande G. Loukanov 1 CUCEI, Unvesdad de Guadalajaa, Apatado Postal 51-71, Col. Las Agulas, C.P , Zapopan, Jalsco, Mexco. CINVESAV, Undad Guadalajaa, Apatado Postal , Plaza La Luna, Guadalajaa, Jalsco, C.P , Mexco. e-mal: almayalans@gmal.com Atcle eceved on Novembe 5, 008; accepted on Mach 3, 009 Abstact. hs pape deals wth the dscete-tme nonlnea system dentfcaton va Recuent Hgh Ode Neual Netwoks, taned wth an extended Kalman flte (EKF based algothm. he pape also ncludes the espectve stablty analyss on the bass of the Lyapunov appoach fo the whole scheme. Applcablty of the scheme s llustated va eal-tme mplementaton fo a thee phase nducton moto. Keywods: Neual dentfcaton, Extended Kalman flteng leanng, Dscete-tme nonlnea systems, hee phase nducton moto. Resumen. Este atículo tata el poblema de dentfcacón de sstemas no lneales dscetos usando edes neuonales ecuentes de alto oden entenadas con un algotmo basado en el flto de Kalman extenddo (EKF. El atículo tambén ncluye el análss de establdad paa el sstema completo, en las bases de la técnca de Lyapunov. La aplcabldad del esquema se lusta a tavés de la mplementacón en tempo eal paa un moto de nduccón tfásco. Palabas clave: Identfcacón neuonal, Apendzaje usando flto de Kalman Extenddo, Sstemas no lneales dscetos, Moto de nduccón tfásco. 1 Intoducton Snce the semnal pape [Naenda and Pathasaathy, 1990], Neual netwoks (NN have become a wellestablshed methodology as exemplfed by the applcatons to dentfcaton and contol of geneal nonlnea and complex systems. In patcula, the use of ecuent hgh ode neual netwoks (RHONN has nceased ecently [Sanchez and Rcalde, 003] due to the excellent appoxmaton capabltes, equng less unts, compaed to the fst ode ones; they ae also moe flexble and obust when faced wth new and/o nosy data pattens [Ghosh and Shn, 199]. Futhemoe, seveal authos have demonstated the feasblty of usng these achtectues n applcatons such as system dentfcaton and contol [Ge, et al., 004; Haykn, 1999; Km and Lews, 1998; Naenda and Pathasaathy, 1990; Rovthaks and Chstodolou, 000; Sanchez, et al., 004; and efeences theen]. hee ae ecent esults whch llustate that the NN technque s hghly effectve n the dentfcaton of a boad categoy of complex dscete-tme nonlnea systems wthout equng complete model nfomaton [Yu and L, 003; Yu and L, 004]. he best well-known tanng appoach fo ecuent neual netwoks (RNN s the back popagaton though tme leanng [Snghal and Wu, 1989]. Howeve, t s a fst ode gadent descent method and hence ts leanng speed could be vey slow [Snghal and Wu, 1989]. Recently the Extended Kalman Flte (EKF based algothms has been ntoduced to tan neual netwoks, n ode to mpove the leanng convegence [Snghal and Wu, 1989]. he EKF tanng of neual netwoks, both feedfowad and ecuent ones, has poven to be elable and pactcal fo many applcatons ove the past ten yeas [Snghal and Wu, 1989]. In [Rovthaks and Chstodolou, 000], adaptve dentfcaton and contol by means of on-lne leanng s analyzed; the stablty of the closed loop system s establshed based on the Lyapunov functon method. Lyapunov appoach can be used dectly to obtan obust tanng algothms fo contnuous-tme ecuent neual netwoks [Sanchez and Rcalde, 003; Rovthaks and Chstodolou, 000]. Fo dscete-tme systems, the poblem s moe complex due to the couplngs among subsystems, nputs and outputs. Few esults have been publshed n compason wth those Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

2 64 Alma Y. Alans, Edga N. Sanchez and Alexande G. Loukanov fo contnuous-tme doman [Yu and L, 003; Yu and L, 004]. By othe hand dscete-tme neual netwoks ae moe convenent fo eal-tme applcatons. Fo many nonlnea systems, t s often dffcult to obtan the accuate and fathful mathematcal models, egadng the physcally complex stuctues and hdden paametes as dscussed n [Chu and Chen, 1998]. heefoe, system dentfcaton becomes mpotant and even necessay befoe contol systems can be consdeed not only fo undestandng and pedctng the behavo of the whole system, but also fo obtanng an effectve contol law. he dentfcaton poblem conssts of choosng an appopate dentfcaton model and adjustng ts paametes accodng to some adaptve law, such that the esponse of the model to an nput sgnal (o class of nput sgnals, appoxmates the esponse of the eal system to the same nput [Rovthaks and Chstodolou, 000]. A challenge poblem fo nonlnea systems dentfcaton s to select a sutable stuctue fo the dentfe, capable of appoxmatng the unknown nonlnea dynamcs. In ths consdeaton, t s notable that ecuent neual netwoks offe the advantage of well appoxmatng a nonlnea system to an abtaly accuate level [Cotte, 1990]. In ths pape, a ecuent hgh ode neual netwok (RHONN s used to dentfy the plant model, unde the assumpton of all the state s avalable fo measuement. he onlne leanng algothm fo the RHONN s mplemented usng an Extended Kalman Flte (EKF. he espectve stablty analyss, on the bass of the Lyapunov appoach, s ncluded fo the poposed scheme. he applcablty of ths scheme s llustated by eal-tme mplementaton fo an electc thee phase nducton moto. Defnton 1. he soluton of (1 s semglobally unfomly ultmately bounded (SGUUB, f fo any Ω, a compact subset of n χ Ω, thee exsts an 0 χ( k R and all ( k 0 ε > and a numbe N, χ ( k < ε fo all k > k0 + N. ( 0 such that In othe wods, the soluton of (1 s sad to be SGUUB f, fo any apo gven (abtaly lage bounded set Ω and any apo gven (abtaly small set Ω 0, whch contans (0,0 as an nteo pont, thee exsts a contol u, such that evey tajectoy of the closed loop system statng fom Ω entes the set Ω 0 = { χ χ < ε} n a fnte tme and emans n t theeafte [Ge, et al., 004]. heoem 1 [Ge, et al., 004]. Let V ( χ be a Lyapunov functon fo the dscete-tme system (1, whch satsfes the followng popetes: γ χ k V χ k γ χ k ( ( ( ( ( ( ( χ( + 1 ( χ( = ( χ( ( k 1 V k V k V k ( ( γ χ + γ ς 3 3 whee ς s a postve constant, γ 1 ( and γ ( stctly nceasng functons, and γ ( nondeceasng functon. hus f ( 0 fo V χ < χ > ς 3 ae s a contnuous, then χ ( k s unfomly ultmately bounded,.e. thee s a tme nstant χ k < ς k < k k, such that ( Mathematcal pelmnaes hough ths pape we use k as the step samplng, k 0 Z +, fo the absolute value, fo the Eucldan nom fo vectos and fo any adequate nom fo matces. Fo moe detals elated to ths secton see [Ge, et al., 004]. Consde a MIMO nonlnea system: χ( k+ 1 = F( χ, uk ( (1 n m whee χ R, u R and nonlnea functon. n m n F R R R s 3 Dscete-tme Recuent Neual Netwoks Let consde the followng dscete-tme ecuent hgh ode neual netwok (RHONN, depcted n Fg.1 whch s descbed as: ( ( ( ( x k+ 1 = wz xk, uk, = 1,, n ( whee x s the state of the -th neuon, L s the espectve numbe of hghe-ode connectons, Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

3 Real-tme Dscete Nonlnea Identfcaton va Recuent Hgh Ode Neual Netwoks 65 { I, 1 I,, I L } s a collecton of non-odeed subsets of { 1,,, n}, n s the state dmenson, w s the espectve on-lne adapted weght vecto, and z x, u( k s gven by ( ( (, ( z x k u k wth d ( j d j ( 1 y z 1 j j I 1 d j ( z y j I j = = z d j( L L y j I L j (3 k beng a nonnegatve nteges, and y s defned as follows: y y 1 S( x1 y S x ( n n = = y u n+ 1 1 y u m n+ m In (4, u [ u1, u,, u m ] neual netwok, and S ( s defned by (4 = s the nput vecto to the ( S x = exp (5 ( x Consde the poblem to appoxmate the geneal dscete-tme nonlnea system (1, by the followng dscete-tme RHONN sees-paallel epesentaton [Rovthaks and Chstodolou, 000]: whee χ * ( k 1 w z ( x, u + = + ε (6 z χ s the -th plant state, ε z s a bounded appoxmaton eo, whch can be educed by nceasng the numbe of the adjustable weghts [Rovthaks and Chstodolou, 000]. Assume that thee exsts deal weghts vecto be mnmzed on a compact set * w such that ε z can L Ωz R. he deal * weght vecto w s an atfcal quantty equed fo analytcal pupose [Rovthaks and Chstodolou, 000]. In geneal, t s assumed that ths vecto exsts and s constant but unknown. Let us defne ts estmate as w and the estmaton eo as * ( = ( w k w w k (7 he estmate w s used fo stablty analyss whch wll be dscussed late. Fg. 1. Schematc epesentaton fo a dscete-tme RHONN 4 he EKF anng Algothm Kalman flteng (KF estmates the state of a lnea system wth addtve state and output whte noses [Chu and Chen, 1998; Gove and Hwang, 199]. Fo KF-based neual netwok tanng, the netwok weghts become the states to be estmated, wth the eo between the neual netwok output and the desed output; ths eo s consdeed as addtve whte nose. Fo dentfcaton, the desed output s nfomaton geneated by the plant; n ths pape, the espectve state. Due to the fact that the neual netwok mappng s nonlnea, an extended Kalman Flteng (EKF-type s equed. he tanng goal s to fnd the optmal weght values that mnmze the pedcton eos (the dffeences between the desed outputs and the neual netwok outputs. he EKF-based NN tanng algothm s descbed by K = P H M w ( k+ 1 = w + ηk e (8 P ( k+ 1 = P K H P + Q = 1,, n wth Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

4 66 Alma Y. Alans, Edga N. Sanchez and Alexande G. Loukanov ( = ( + ( ( ( e χ x M k R k H k P k H k (9 = (10 whee e ( P L L at step k, 1 k s the espectve dentfcaton eo, R s the pedcton eo covaance matx L w R s the weght (state vecto, L s the espectve numbe of neual netwok weghts, χ s the -th plant state, x s the -th neual netwok state, n s the numbe of states, Q L K R s the Kalman gan L L vecto, R s the NN weght estmaton nose covaance matx, covaance; R R s the measuement nose L H R s a vecto, n whch each enty ( H j s the devatve of one of the neual netwok state, ( x, wth espect to one neual netwok weght, ( w j, as follows H j x j = w w = w ( k+ 1 (11 whee = 1,,, n and j = 1,,, L. Usually P and Q ae ntalzed as dagonal matces, wth entes 0 Q 0, espectvely. It s mpotant to P ( and ( emak that ( H k, K ( k and ( P k fo the EKF ae bounded; fo a detaled explanaton of ths fact see [Song and Gzzle, 1995]. hen the dynamcs of the dentfcaton eo (10 can be expessed as ( 1 ( ( (, ( e k + = w k z x k u k + ε (1 z By the othe hand the dynamcs of (7 s ( + 1 = ( η ( ( w k w k K k e k (13 Now, we establsh the man esult of ths pape n the followng theoem. heoem : he RHONN ( taned wth the EKFbased algothm (8 to dentfy the nonlnea plant (1, ensues that the dentfcaton eo (10 s semglobally unfomly ultmately bounded (SGUUB; moeove, the RHONN weghts eman bounded. Poof: Consde the Lyapunov functon canddate ( = ( ( ( + ( ( ( 1 ( = w ( k+ 1 P ( k+ 1 w ( k+ 1 w P w + e ( k+ 1 e V k w k P k w k e k V k = V k + V k Usng (1 and (13 n (14 ( ( η ( ( P A w η K e V k = w k K k e k ( ( (, ( ( ( ( ( + w k z x k u k w k P k w k e k wth A K H P Q = + ; then, (15 can be expessed as ( ( ( ( ηe K P w w A w + ηe K A w ηe w P K + η e K P K + ηe w A K η e K A K + ( w z ( x, u + w z ( x, u ε + w P w e V k = w k P k w k mn z z Usng the nequaltes X X + Y Y X Y X X + Y Y X Y ( λ ( λ P X X PX P X max ε (14 (15 (16 n n n whch ae vald XY, R, P R, P= P > 0, then (16, can be ewtten as Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

5 Real-tme Dscete Nonlnea Identfcaton va Recuent Hgh Ode Neual Netwoks 67 ( ( ( ( η e K A K + w w + e + η e K P P K + η w A K K A w + η e K P K + w z( x u + ε z e k V k w k A k w k ï hen ( (, ( ( ( λmn ( ( + η e K λmax ( P + η w λmax ( A K + η e K λmax ( P η e K λmn ( A + w z ( x u + V k w k w k A k ε z, hen, thee exsts η, and F > 0, wth Q and (17 (18 R such that E > 0 ( = λmn ( ( ηλmax ( ( ( z ( x, u 1 ( = η ( λmn ( ( η K λmax ( P = ε E k A k A k K k F k K k A k G z he neual dentfcaton s pefomed on-lne, usng a sees-paallel confguaton as llustated n Fg.. 5 Applcaton Fg.. Neual Identfe scheme In ths secton Real-me esults ae pesented fo the fo neual netwok dentfcaton scheme poposed above. he expements ae pefomed usng a benchmak, whch ncludes: Compute Staton. A PC fo supevson, wth a DS1104 stand alone boad fo data acquston and contol, and the equed softwae (Fg. 3. Sensos. One encode, cuent sensos, and L to CMOS couplng (Fg. 4. Electonc Powe Staton. A thee-phase dve, wth the equed IGBs (Fg. 5. Benchmak. A thee-phase squel cage nducton moto (Fg. 4. It s mpotant to emak that the nducton moto paametes ae unknown. heefoe, (18 can be expessed as ( ( ( ( ( ( V k w k E k e k F k + G k hen V 0 < when ( ( ( ( G k G k w κ e E k F k 1 OR κ heefoe, accodng to heoem 1, the soluton of (1 and (13 s stable, hence the dentfcaton eo and the RHONN weghts ae SGUUB. Fg. 3. Vew of the PC and the DS1104 boad Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

6 68 Alma Y. Alans, Edga N. Sanchez and Alexande G. Loukanov Fg. 4. Encode coupled wth the nducton moto 5.1 Moto model Fg. 5. PWM dve he sx-ode dscete-tme nducton moto model n,, unde the the stato fxed efeence fame ( assumptons of equal mutual nductances and lnea magnetc ccut, s gven by [Loukanov, et al., 00] µ ω ω ( ψ ψ L J ( k+ 1 = + ( 1 a M ( p 1 ( ( npθ ( k ρ ( p 1 ( ( npθ ( k ρ ( + 1 = cos ( + 1 ψ k n θ k ρ k sn + 1 ( + 1 = sn ( + 1 ψ k n θ k ρ k + cos + 1 ( k + 1 = ϕ + u σ ( k + 1 = ϕ + u σ L θ ( k + 1 = θ + ω J wth 1 µ + ( 1 a ( ( ψ ( ( ψ ( M k k k k ( = a cos( Φ + sn ( Φ + b( cos( Φ + sn ( Φ = a cos( Φ sn ( Φ + b( cos( Φ sn ( Φ = + + np γ ( k = + + np γ ( k ρ ψ ψ ( ρ ψ ψ ϕ ψ ω ψ ϕ ψ ω ψ (19 (0 wth b ( 1 R = a M, =, L n θ Φ =, and p Mn p µ =, besdes s JL γ M M σ = Ls, =, L σ L L, R = M R s σl + σ, a = e L and M ae the stato, oto and mutual nductance espectvely; R s and ae the stato and oto esstances espectvely; n s p R Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

7 Real-tme Dscete Nonlnea Identfcaton va Recuent Hgh Ode Neual Netwoks 69 the numbe of pole pas; and epesents the cuents n the and phases, espectvely; ψ and ψ epesents the fluxes n the and phases, espectvely and θ s the oto angula dsplacement. 5. Neual netwok dentfcaton he RHONN poposed fo ths applcaton s as follows: x1 ( k+ 1 = w11 ( ks ( ω + w1 ( ks ( ω S( ψ + w13 ( ks ( ω S( ψ x ( k+ 1 = w1( ks ( ω S( ψ + w x3 ( k+ 1 = w31 ( ks ( ω S( ψ + w3 (1 x4 ( k+ 1 = w41( ks ( ψ + w4 ( ks ( ψ + w43 ( ks ( + w44u x5 ( k+ 1 = w51( ks ( ψ + w5 ( ks ( ψ + w ( ks ( + w u he tanng s pefomed on-lne, usng a seespaallel confguaton as llustated n Fg.. Dung the dentfcaton pocess the plant and the NN opeates n open-loop. Both of them (plant and NN have the same nput vecto u, u ; All the NN states ae ntalzed n a andom way as well as the weghts vectos. It s mpotant emak that the ntal condtons of the plant ae completely dffeent fom the ntal condtons fo the NN. he dentfcaton s pefomed usng (8 wth = 1,,, n wth n the dmenson of plant state ( n = Real-tme esults In ths subsecton the neual netwok dentfcaton scheme poposed above fo the dscete-tme nducton moto model s appled n eal-tme to the descbed benchmak. Dung the dentfcaton pocess the plant and the NN opeates n open-loop. Both of them (plant and NN have the same nput vecto u, u ; u and u ae chp functons wth 00 volts of ampltude and ncemental fequences fom 0 Hz to 150 Hz and 0 Hz to 00 Hz, espectvely. he mplementaton s pefomed wth a samplng tme of s. he esults of the eal-tme mplementaton ae pesented as follows: Fg. 6 dsplays the dentfcaton pefomance fo the speed oto, plant sgnal s n sold lne and neual sgnal s n dashed one, the ovelap s due to the excellent pefomance of the neual dentfe, the standad devaton fo the dentfcaton eo ω x1 s ad / s ; Fg. 7 and Fg. 8 pesent the dentfcaton pefomance fo the fluxes n phase and espectvely, plant sgnal s n sold lne and neual sgnal s n dashed one, the ovelap s due to the excellent pefomance of the neual dentfe, the standad devaton fo flux dentfcaton eos ψ x and ψ x3 ae 0.044wb and 0.063wb, espectvely;. Fg. 9 and Fg. 10 potay the dentfcaton pefomance fo cuents n phase and espectvely plant sgnal s n sold lne and neual sgnal s n dashed one, the ovelap s due to the excellent pefomance of the neual dentfe, the standad devaton fo cuent dentfcaton eos x and 4 x ae A and A 5 espectvely. Fnally the nput sgnals ae pesented n Fg. 11. Fg. 6. Real tme oto speed dentfcaton (plant sgnal n sold lne and neual sgnal n dashed lne Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

8 70 Alma Y. Alans, Edga N. Sanchez and Alexande G. Loukanov Fg. 7. Real tme alpha flux dentfcaton (plant sgnal n sold lne and neual sgnal n dashed lne Fg. 10. Real tme beta cuent speed dentfcaton (plant sgnal n sold lne and neual sgnal n dashed lne Fg. 8. Real tme oto beta flux dentfcaton (plant sgnal n sold lne and neual sgnal n dashed lne Fg. 11. Input sgnals appled dung the dentfcaton u k u k n dashed lne pocess ( ( n sold lne and ( 6 Conclusons Fg. 9. Real tme oto alpha cuent dentfcaton (plant sgnal n sold lne and neual sgnal n dashed lne hs pape has pesented the applcaton of ecuent hgh ode neual netwoks to dentfcaton of dscetetme nonlnea systems. he tanng of the neual netwoks was pefomed on-lne usng an extended Kalman flte. he boundness of the dentfcaton eo was establshed on the bass of the Lyapunov appoach. he RHONN tanng wth the EKF-based algothm, pesents good pefomance. Real-tme esults show the effectveness of the poposed schemes, as appled to an electc thee-phase squel cage nducton moto. hs pape deals only wth onlne dentfcaton fo a thee phase nducton moto, Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

9 Real-tme Dscete Nonlnea Identfcaton va Recuent Hgh Ode Neual Netwoks 71 contol synthess and mplementaton based on the poposed appoaches s consdeed as futue wok. Acknowledgement: he authos thank the suppot of PROMEP/103.5/09/391 and CONACY Mexco, though Poject Y. hey also thank the vey useful comments of the anonymous evewes, whch help to mpove the pape. Refeences Chu, C. K., & Chen, G. (1998. Kalman Flteng wth Real-me Applcatons. New Yok: Spnge- Velag. Cotte, N. E. (1990. he Stone-Weetass theoem and ts applcaton to neual netwoks. IEEE ansactons Neual Netwoks. 1(4, Ge, S. S., Zhang, J. & Lee,. H. (004. Adaptve neual netwok contol fo a class of MIMO nonlnea systems wth dstubances n dscetetme. IEEE ansactons on Systems, Man and Cybenetcs, Pat B, 34(4, Ghosh, J. & Shn Y. (199. Effcent Hgh-Ode Neual Netwoks fo Classfcaton and Functon Appoxmaton. Intenatonal. Jounal of Neual Systems, 3(4, Gove, R., & Hwang, P. Y. C. (199. Intoducton to Random Sgnals and Appled Kalman Flteng. New Yok: John Wley and Sons. Haykn, S. (1999. Neual Netwoks. A compehensve foundaton. New Jesey: Pentce Hall. Km, Y. H., & Lews, F. L. (1998. Hgh-Level Feedback Contol wth Neual Netwoks. Sngapoe: Wold Scentfc. Loukanov, A. G., Rvea, J. & Cañedo J. M. (00. Dscete-tme sldng mode contol of an nducton moto. 00 IFAC 15th ennal Wold Congess, Bacelone, Span, Naenda, K. S., & Pathasaathy, K. (1990. Identfcaton and contol of dynamcal systems usng neual netwoks. IEEE ansactons on Neual Netwoks, 1(1, 4-7. Rovthaks, G. A., & Chstodoulou, M. A. (000. Adaptve Contol wth Recuent Hgh -Ode Neual Netwoks. New Yok: Spnge Velag. Sanchez, E. N., Alans, A. Y. & Chen, G. (004. Recuent neual netwoks taned wth Kalman flteng fo dscete chaos econstucton. Asan- Pacfc Wokshop on Chaos Contol and Synchonzaton 04, Melboune, Austala, Sanchez, E. N., & Rcalde, L. J. (003. ajectoy tackng va adaptve ecuent neual contol wth nput satuaton. Intenatonal Jont Confeence on Neual Netwoks 03, Potland, USA, vol.1, Snghal, S., & Wu, L. (1989. anng multlaye peceptons wth the extended Kalman algothm. In D. S. ouetzky (Ed., Advances n Neual Infomaton Pocessng Systems ( San Mateo, CA: Mogan Kaufmann. Song, Y., & Gzzle, J. W. (1995. he extended Kalman Flte as Local Asymptotc Obseve fo Dscete-me Nonlnea Systems. Jounal of Mathematcal systems, Estmaton and Contol, 5(1, Yu, W., & L, X. (003. Dscete-tme neuo dentfcaton wthout obust modfcaton. IEE Poceedngs Contol heoy & Applcatons, 150(3, Yu, W. (004. Nonlnea system dentfcaton usng dscete-tme ecuent neual netwoks wth stable leanng algothms. Infomaton Scences Infomatcs and Compute Scence: An Intenatonal Jounal, 158 (1, Alma Y. Alans was bon n Duango, Duango, Mexco, n She eceved the B. Sc degee fom Insttuto ecnologco de Duango (ID, Duango, Duango, n 00, the M.Sc. and the Ph.D. degees n electcal engneeng fom the Advanced Studes and Reseach Cente of the Natonal Polytechnc Insttute (CINVESAV-IPN, Guadalajaa Campus, Mexco, n 004 and 007, espectvely. Snce 008 she has been wth Unvesty of Guadalajaa, whee she s cuently a Pofesso n the Depatment of Compute Scence. She s also membe of the Mexcan Natonal Reseach System (SNI-1. He eseach nteest centes on neual contol and modelng fo nonlnea systems. Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7

10 7 Alma Y. Alans, Edga N. Sanchez and Alexande G. Loukanov Edga N. Sanchez was bon n 1949, n Sadnata, Colomba, South Ameca. He obtaned the BSEE, majo n Powe Systems, fom Unvesdad Industal de Santande (UIS, Bucaamanga, Colomba n 1971, the MSEE fom CINVESAV-IPN (Advanced Studes and Reseach Cente of the Natonal Polytechnc Insttute, majo n Automatc Contol, Mexco Cty, Mexco, n 1974 and the Docteu Ingeneu degee n Automatc Contol fom Insttut Natonale Polytechnque de Genoble, Fance n He s also membe of the Mexcan Natonal Reseach System (pomoted to hghest ank, III, n 005, the Mexcan Academy of Scence and the Mexcan Academy of Engneeng. Alexande G. Loukanov was bon n 1946, n Moscow, Russa. He gaduated fom Polytechnc Insttute, (Dpl. Eng., Moscow, Russa n 1975, and eceved Ph. D. n Automatc Contol fom Insttute of Contol Scences of Russan Academy of Scences, Moscow, Russa n He has been wth CINVESAV-IPN, (Advanced Studes and Reseach Cente of the Natonal Polytechnc Insttute, Guadalajaa Campus, Mexco, as a Pofesso of Electcal Engneeng gaduate pogams. Hs eseach nteest cente n Nonlnea Systems Contol and Vaable Stuctue Systems wth Sldng Mode as appled to electc dves and powe systems contol, space and automotve contol. Computacón y Sstemas Vol. 14 No. 1, 010, pp 63-7